Obtaining Net Work

After my first meeting with Mr.H I was stumped on how Net Work was obtained in the Stirling Engine. If the law of conservation of energy was to be true wouldn’t the work required to push the pistons upwards equal that of the work required to move the piston downwards. After some thinking I realized I was missing a key idea; fluctuations in the pressure of the main cylinder change the forces acting on the piston. Using this I was able to reason how Net Work could be obtained. For this post the only new reading I completed was to investigate how to calculate the force exerted by a gas in a closed system and what a PV diagram shows.

The PV diagram to the right shows Pressure and Volume fluctuations within one cycle of the Stirling Engine.

Four unique phases are shown in this idealized case. Realistically the diagram looks closer to the shape a banana, however the only changes between the idealized case and the realistic diagram is that the four phases are more segregated. Meaning in reality the four phases still exist but the drive mechanisms result in the four phases overlapping, slightly changing the overall appearance of the diagram. The four phases correspond with the motion of the displacer and working piston. When pressure increases or decreases the displacer is moving, changing the heat source and resulting in higher or lower pressures (phase 4 and 2 in the diagram). The motion of the piston is illustrated in phases 4 and 2.

Since Work only occurs when force is parallel to displacement, the motion of the displacer can be neglected because there are no changes to the volume of the system during its motion. As a result to calculate the Net Work of the Engine only the motion of the working piston needs to be analyzed. From looking at my model of the Engine I realized the piston moves in two circumstances:

1.  The piston pushes upwards when there is high pressure in the cylinder
2. The piston falls when there is low pressure in the cylinder

Since I needed to find the work of the piston in each circumstance I started to adapt the equation W = F_||∆d. I went to the internet and found that the force a gas exerts on an object can by explained through the equation F­­_gas = P x A, where P is the pressure in Pascal’s and A is the area of the surface the gas is pushing against. I checked the equation to make sure that this equation was an accurate representation of the force applied by the gas:

F­­_gas = P x A

N = N/m^{2 x} m²

N = N

This shows that the force of the gas can be explained by the pressure of the gas and the area of the surface the gas is pushing against. Furthermore, I reasoned that since the system is in 3 dimensions the displacement of the piston should be described by volume. Since the volume of a cylinder is the area of its circular shape multiplied by the length volume can be shown in the equation V = A x ∆d. Rearranged I found that ∆d = V/A. I then substituted these new equations into the work equation:

W = F x d

W = (PA)(V/A)

W = PV

Thusly the Work done at any point by the working piston is the product of pressure and a change in volume. Returning to the PV diagram, it is seen that during phase 1 and 3 the change in volumes are equivalent, but during phase 1 the pressure is much higher than the pressure in phase 3. If the work is equal to the product of pressure and the changes in volume it is easily distinguishable that the work done during phase 1 will be larger than the work done during phase 3. As a result the Net Work of the Engine is the work done in phase 1 minus the work done in phase 3.

As I mentioned earlier little reading helped with my understanding of how Net Work occurred within the Stirling Engine. Beyond stating that the PV diagram shows the Work done by the Engine and that the motion of the displacer need not be analyzed my textbook gave no information. Likewise the internet was used to find the units of measure for a Pascal and the force the gas exerted on the piston. My understanding of Net Work in the Stirling Engine was through slowly going through the above equations aided by my own reasoning and prior knowledge.  

Sources:

Hooper, C & Reader, T. G. (1983). STIRLING ENGINES. New York, NY: E. & F. N. Spon.

Wikipedia. (May 26 2014). Pressure. Retrieved May 28 2014 from “Wikipedia”: www.wikipedia.org